Mr. Ivory on the Method of the Least Squares. 165 



£ + S' + S" + &C. = 0, 



which is equivalent to the rule of the arithmetical mean. In 

 every other system the errors have a constant part common 

 to them all, as is evident from what the general expressions 

 become in the particular case under consideration, viz. 



e = e + (<? — e) 



e' = * 

 e" = 



' + (e - 

 e"+ (e-.O 



&c. 



Such a set of errors cannot occur unless by the operation of 

 some preponderating cause tending to augment or diminish 

 every error in an equal degree. If there be no bias in the 

 experiments, we must necessarily adopt the only system in 

 which every error is singly determined by its proper quantities 

 independently of all the other errors. The rule of the arith- 

 metical mean is therefore only a particular inference from the 

 general reasoning we have been explaining. 



On account of its great practical utility, the method of the 

 least squares has been the subject of much discussion on the 

 continent. In order to prove it recourse has been had to the 

 doctrine of probabilities and the most abstruse researches of 

 the algebraic calculus. But the real grounds of the method 

 are undoubtedly, the general character of the errors, and the 

 properties of the equations of condition. From this last con- 

 sideration we readily deduce that there is only one system of 

 errors, or perhaps in particular circumstances, only a certain 

 number of systems, that can possibly be consistent with the 

 nature of the experiments. Viewed in this light, the method 

 of the least squares is separated from the laws of chance, and 

 is made to depend on very simple and elementary principles. 

 The application of probability to physical researches rests on 

 other grounds ; namely, the general expression of the chance 

 of an error so modified as to lead to rules sufficiently simple 

 for application in practice. Although I have here confined 

 myself to the most simple case of only one correction, yet the 

 principles laid down are general, and are readily applied to 

 several independent corrections by a repetition of similar rea- 

 soning. 



Sept. 4, 1826. J. Ivory. 



XXIIL On 



